Editing Differential operator (section)

==Coordinate-independent description==
In [[differential geometry]] and [[algebraic geometry]] it is often convenient to have a [[coordinate]]-independent description of differential operators between two [[vector bundle]]s.  Let ''E'' and ''F'' be two vector bundles over a [[differentiable manifold]] ''M''. An '''R'''-linear mapping of [[vector bundle|sections]] {{nowrap|''P'' : Γ(''E'') → Γ(''F'')}} is said to be a '''''k''th-order linear differential operator''' if it factors through the [[jet bundle]] ''J''<sup>''k''</sup>(''E'').
In other words, there exists a linear mapping of vector bundles

:<math>i_P: J^k(E) \to F</math>

such that

:<math>P = i_P\circ j^k</math>

where {{nowrap|''j''<sup>''k''</sup>: Γ(''E'') → Γ(''J''<sup>''k''</sup>(''E''))}} is the prolongation that associates to any section of ''E'' its [[jet (mathematics)|''k''-jet]].

This just means that for a given [[vector bundle|section]] ''s'' of ''E'', the value of ''P''(''s'') at a point ''x''&nbsp;∈&nbsp;''M'' is fully determined by the ''k''th-order infinitesimal behavior of ''s'' in ''x''. In particular this implies that ''P''(''s'')(''x'') is determined by the [[sheaf (mathematics)|germ]] of ''s'' in ''x'', which is expressed by saying that differential operators are local. A foundational result is the [[Peetre theorem]] showing that the converse is also true: any (linear) local operator is differential.

===Relation to commutative algebra===
{{Main Article|Differential calculus over commutative algebras}}

An equivalent, but purely algebraic description of linear differential operators is as follows: an '''R'''-linear map ''P'' is a ''k''th-order linear differential operator, if for any ''k''&nbsp;+&thinsp;1 smooth functions <math>f_0,\ldots,f_k \in C^\infty(M)</math> we have

:<math>[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.</math>

Here the bracket <math>[f,P]:\Gamma(E)\to \Gamma(F)</math> is defined as the commutator

:<math>[f,P](s)=P(f\cdot s)-f\cdot P(s).</math>

This characterization of linear differential operators shows that they are particular mappings between [[module (mathematics)|modules]] over a [[commutative algebra (structure)|commutative algebra]], allowing the concept to be seen as a part of [[commutative algebra]].